Method of performing tomographic imaging in a charged-particle microscope

ABSTRACT

A method of performing sub-surface imaging of a specimen in a charged-particle microscope of a scanning transmission type, comprising the following steps:
         Providing a beam of charged particles that is directed from a source along a particle-optical axis through an illuminator so as to irradiate the specimen;   Providing a detector for detecting a flux of charged particles traversing the specimen;   Causing said beam to follow a scan path across a surface of said specimen, and recording an output of said detector as a function of scan position, thereby acquiring a scanned charged-particle image I of the specimen;   Repeating this procedure for different members n of an integer sequence, by choosing a value P n  of a variable beam parameter P and acquiring an associated scanned image I n , thereby compiling a measurement set M={(I n , P n )};   Using computer processing apparatus to automatically deconvolve the measurement set M and spatially resolve it into a result set representing depth-resolved imagery of the specimen,
 
wherein:
   Said variable beam parameter P is focus position along said particle-optical axis;   Said scanned image I is an integrated vector field image, obtained by;
           Embodying said detector to comprise a plurality of detection segments;   Combining signals from different detection segments so as to produce a vector output from the detector at each scan position, and compiling this data to yield a vector field;   Mathematically processing said vector field by subjecting it to a two-dimensional integration operation.

The invention relates to a method of performing sub-surface imaging of aspecimen in a charged-particle microscope of a scanning transmissiontype, comprising the following steps:

-   -   Providing a beam of charged particles that is directed from a        source along a particle-optical axis through an illuminator so        as to irradiate the specimen;    -   Providing a detector for detecting a flux of charged particles        traversing the specimen;    -   Causing said beam to follow a scan path across a surface of said        specimen, and recording an output of said detector as a function        of scan position, thereby acquiring a scanned charged-particle        image I of the specimen;    -   Repeating this procedure for different members n of an integer        sequence, by choosing a value P_(n) of a variable beam parameter        P and acquiring an associated scanned image In, thereby        compiling a measurement set M={(I_(n), P_(n))};    -   Using computer processing apparatus to automatically deconvolve        the measurement set M and spatially resolve it into a result set        representing depth-resolved imagery of the specimen.

The invention also relates to a charged-particle microscope in whichsuch a method can be performed.

Charged-particle microscopy is a well-known and increasingly importanttechnique for imaging microscopic objects, particularly in the form ofelectron microscopy. Historically, the basic genus of electronmicroscope has undergone evolution into a number of well-known apparatusspecies, such as the Transmission Electron Microscope (TEM), ScanningElectron Microscope (SEM), and Scanning Transmission Electron Microscope(STEM), and also into various sub-species, such as so-called “dual-beam”tools (e.g. a FIB-SEM), which additionally employ a “machining” FocusedIon Beam (FIB), allowing supportive activities such as ion-beam millingor Ion-Beam-Induced Deposition (IBID), for example. More specifically:

-   -   In a SEM, irradiation of a specimen by a scanning electron beam        precipitates emanation of “auxiliary” radiation from the        specimen, in the form of secondary electrons, backscattered        electrons, X-rays and photoluminescence (infrared, visible        and/or ultraviolet photons), for example; one or more components        of this emanating radiation is/are then detected and used for        image accumulation purposes, and/or spectroscopic analysis (as        in the case of EDX (Energy-Dispersive X-Ray Spectroscopy), for        example).    -   In a TEM, the electron beam used to irradiate the specimen is        chosen to be of a high-enough energy to penetrate the specimen        (which, to this end, will generally be thinner than in the case        of a SEM specimen); the flux of transmitted electrons emanating        from the specimen can then be used to create an image, or        produce a spectrum (as in the case of EELS, for example;        EELS=Electron Energy-Loss Spectroscopy). If such a TEM is        operated in scanning mode (thus becoming a STEM), the        image/spectrum in question will be accumulated during a scanning        motion of the irradiating electron beam.        More information on some of the topics elucidated here can, for        example, be gleaned from the following Wikipedia links:    -   http://en.wikipedia.org/wiki/Electron_microscope    -   http:/en.wikipedia.org/wiki/Scanning_electron_microscope    -   http://en.wikipedia.org/wiki/Transmission_electron_microscopy    -   http://en.wikipedia.org/wiki/Scanning_transmission_electron_microscopy        As an alternative to the use of electrons as irradiating beam,        charged-particle microscopy can also be performed using other        species of charged particle. In this respect, the phrase        “charged particle” should be broadly interpreted as encompassing        electrons, positive ions (e.g. Ga or He ions), negative ions,        protons and positrons, for instance. As regards ion-based        microscopy, some further information can, for example, be        gleaned from sources such as the following:    -   http://en.wikipedia.org/wiki/Scanning_Helium_Ion_Microscope    -   W. H. Escovitz, T. R. Fox and R. Levi-Setti, Scanning        Transmission Ion Microscope with a Field Ion Source, Proc. Nat.        Acad. Sci. USA 72(5), pp 1826-1828 (1975).    -   http://www.innovationmagazine.com/innovation/volumes/v7n1/coverstory3.shtml        It should be noted that, in addition to imaging and/or        spectroscopy, a charged-particle microscope (CPM) may also have        other functionalities, such as examining diffractograms,        performing (localized) surface modification (e.g. milling,        etching, deposition), etc.

In all cases, a Scanning Transmission Charged-Particle Microscope(STCPM) will comprise at least the following components:

-   -   A radiation source, such as a Schottky electron source or ion        gun.    -   An illuminator, which serves to manipulate a “raw” radiation        beam from the source and perform upon it certain operations such        as focusing, aberration mitigation, cropping (with a        stop/iris/condensing aperture), filtering, etc. It will        generally comprise one or more charged-particle lenses, and may        comprise other types of particle-optical component also.    -   If desired, the illuminator can be provided with a deflector        system that can be invoked to cause its output beam to perform a        scanning motion across the specimen being investigated.    -   A specimen holder, on which a specimen under investigation can        be held and positioned (e.g. tilted, rotated). If desired, this        holder can be moved so as to effect a scanning motion of the        beam w.r.t. the specimen. In general, such a specimen holder        will be connected to a positioning system such as a mechanical        stage.    -   An imaging system, which essentially takes charged particles        that are transmitted through a specimen (plane) and directs        (focuses) them onto analysis apparatus, such as a        detection/imaging device, spectroscopic apparatus, etc. As with        the illuminator referred to above, the imaging system may also        perform other functions, such as aberration mitigation,        cropping, filtering, etc., and it will generally comprise one or        more charged-particle lenses and/or other types of        particle-optical components.    -   A detector, which may be unitary or compound/distributed in        nature, and which can take many different forms, depending on        the radiation/entity being recorded. Such a detector may, for        example, be used to register an intensity value, to capture an        image, or to record a spectrum. Examples include        photomultipliers (including solid-state photomultipliers,        SSPMs), photodiodes, (pixelated) CMOS detectors, (pixelated) CCD        detectors, photovoltaic cells, etc., which may, for example, be        used in conjunction with a scintillator film, for instance. For        X-ray detection, use is typically made of a so-called Silicon        Drift Detector (SDD), or a Silicon Lithium (Si(Li)) detector,        for example. Typically, an STCPM will comprise several        detectors, of various types.        In what follows, the invention may—by way of example—sometimes        be set forth in the specific context of electron microscopy.        However, such simplification is intended solely for        clarity/illustrative purposes, and should not be interpreted as        limiting.

An example of a method as set forth in the opening paragraph above isknown from so-called HAADF-STEM tomography (HAADF=High-Angle AnnularDark Field), in which the beam parameter P is beam incidence angle (beamtilt) relative to (a plane of) the specimen, and in which themeasurement set M is a so-called “tilt series” or “sinogram”. See, forexample, the following publication:

-   -   C. Kübel et al., Recent advances in electron tomography: TEM and        HAADF-STEM tomography for materials science and semiconductor        applications, Microscopy and Microanalysis November 2005, pp.        378-400:    -   http//www.researchgate.net/publication/6349887_Recent_advances_in_electron_tomogra        phy_TEM_and_HAADF-STEM_tomography_for_materials_science_and_semiconductor_applications        In this known technique, the deconvolution/spatial resolution        (“reconstruction”) of the set M can be performed using various        mathematical tools. For example:    -   SIRT: Simultaneous Iterative Reconstruction Technique.        -   See, for example:            -   http://www.vcipt.org/pdfs/wcipt1/s2_1.pdf            -   P. Gilbert, Journal of Theoretical Biology, Volume 36,                Issue 1, July 1972, Pages 105-117.        -   DART: Discrete Algebraic reconstruction Technique.            -   See, for example:                http://en.wikipedia.org/wiki/Algebraic_reconstruction_technique                http://www.emat.ua.ac.be/pdf/1701.pdf. and references                therein.        -   FST: Fourier Slice Theorem.            -   See, for example, the book by A. C. Kak and Malcolm                Slaney, Principles of Computerized Tomographic Imaging,                IEEE Press, 1999; in particular, chapter 3, especially                sections 3.2 and 3.3.        -   WBP (Weighted Back Projection) and POCS (Projection Onto            Convex Sets), etc.

Although prior-art techniques such as set forth in the previousparagraph have produced tolerable results up to now, the currentinventors have worked extensively to provide an innovative alternativeto the conventional approach. The results of this endeavor are thesubject of the current invention.

It is an object of the invention to provide a radically new method ofinvestigating a specimen using an STCPM. In particular, it is an objectof the invention that this method should allow sub-surface imaging ofthe specimen using alternative acquisition and processing techniques tothose currently used.

These and other objects are achieved in a method as set forth in theopening paragraph above, which method is characterized in that:

-   -   Said variable beam parameter P is focus position (F) along said        particle-optical axis;    -   Said scanned image I is an integrated vector field image,        obtained by;        -   Embodying said detector to comprise a plurality of detection            segments;        -   Combining signals from different detection segments so as to            produce a vector output from the detector at each scan            position, and compiling this data to yield a vector field;        -   Mathematically processing said vector field by subjecting it            to a two-dimensional integration operation.

The current invention makes use of integrated vector field (iVF)imaging, which is an innovative imaging technique set forth inco-pending European Patent Applications EP 14156356 and EP 15156053, andco-pending U.S. patent application U.S. Ser. No. 14/629,387 (filed Feb.23, 2015)—which are incorporated herein by reference, and will bereferred to hereunder as the “iVF documents”. Apart from this differencein the nature of the employed image I, the invention also differs fromthe prior art in that the adjusted beam parameter P (which is(incrementally) changed so as to obtain the measurement set M) is axialfocus position instead of beam tilt. Inter alia as a result of thesedifferences—and various insights attendant thereto, which will beelucidated in greater detail below—the invention can make use of adifferent mathematical approach to perform depth-resolution on themeasurement set M.

Of significant importance in the present invention is the insight thatan iVF image is essentially a map of electrostatic potential φ(x, y) inthe specimen, whereas a HAADF-STEM image is a map of φ²(x,y) [see nextparagraph also]. As a result of this key distinction, one can make useof a linear imaging model (and associated deconvolution techniques) inthe present invention, whereas one cannot do this when using HAADF-STEMimagery. More specifically, this linearity allows imagecomposition/deconvolution to be mathematically treated as a so-calledSource Separation (SS) problem (e.g. a Blind Source Separation (BSS)problem), in which an acquired image is regarded as being a convolutionof contributions from a collection of sub-sources distributed within thebulk of the specimen, and in which sub-source recovery can be achievedusing a so-called Inverse Problem Solver; this contrasts significantlywith HAADF-STEM tomography, in which image reconstruction is based onmathematics that rely on “line-of-sight” or “parallax” principles (basedon so-called Radon Transforms). The SS approach is made possible becausethe aforementioned linearity implies minimal/negligible interferencebetween said sub-sources, and it preserves phase/sign—which is lost whenworking with a quadratic entity such as φ²(x,y)[as in HAADF-STEM].Moreover, the irradiating charged-particle beam in an STCPM caneffectively be regarded as passing directly through the specimen, withnegligible lateral spread (scattering); as a result, there will berelatively low loss of lateral resolution in an associated SS problem.Examples of mathematical techniques that can be used to solve an SSproblem as alluded to here include, for example, Principal ComponentAnalysis (PCA), Independent Component Analysis (ICA), Singular ValueDecomposition (SVD) and Positive Matrix Formulation (PMF). Moreinformation with regard to SS techniques can, for example, be gleanedfrom:

-   -   [1] P. Comon and C. Jutten, Handbook of Blind Source Separation:        Independent Component Analysis and Applications, Academic Press,        2010.    -   [2] A. Hyvarinen and E. Oja, Independent Component Analysis:        Algorithms and Applications, Neural Networks, 13(4-5):411-430,        2000.    -   [3] I. T. Jolliffe, Principal Component Analysis, Series:        Springer Series in Statistics XXIX, 2nd ed., Springer, N Y,        2002.

At this point, it should be noted that the current invention issubstantially different from the technique commonly referred to as“Confocal STEM” or “SCEM” (SCEM=Scanning Confocal Electron Microscopy).In this known technique:

-   -   One is not performing any imaging deconvolution. Instead, one is        assuming that, for a given focus position within a specimen, all        imaging information is produced within the corresponding focal        plane (so-called “waist” of the incoming focused        charged-particle beam/probe), with no significant contributions        from overlying or underlying layers. Consequently, for each        employed focus setting, there is no        mixing/degeneration/convolution of imaging information from a        stack of co-contributing layers, and thus no associated        deconvolution/disentanglement problem.    -   Considering a single layer of the specimen (thinner than the        beam waist), one is not imaging φ(x, y) (since one is not using        iVF imagery), but one is instead imaging a complicated        non-linear function of φ(x, y). More specifically, the image        formation process in SCEM is given by the expression:

I ^(SCEM)({right arrow over (r)} _(p))=|ψ_(in) ^(L1)(−{right arrow over(r)})ψ_(in) ^(L2)({right arrow over (r)})*e^(iφ({right arrow over (r)}))({right arrow over (r)} _(p))|²

-   -   wherein:        -   ψ_(in) ^(L1)(−{right arrow over (r)}) and ψ_(in)            ^(L2)({right arrow over (r)}) are wave functions            respectively associated with the condenser lens            (illuminator) and projection lens (imaging system);        -   ψ_(in) ^(L1)(−{right arrow over (r)}) describes the probe            (irradiating charged-particle beam) impinging upon the            specimen;        -   I represents intensity and {right arrow over (r)}_(p)            denotes scanning coordinate/probe position;        -   The “*” operator indicates a convolution.    -   In contrast, in the present invention (with iVF imaging), one        instead obtains:

${I^{iVF}\left( {\overset{\rightarrow}{r}}_{p} \right)} = {\frac{1}{2\pi}\left( {{{\psi_{in}\left( \overset{\rightarrow}{r} \right)}}^{2}{{\bigstar\phi}\left( \overset{\rightarrow}{r} \right)}} \right)\left( {\overset{\rightarrow}{r}}_{p} \right)}$

-   -   in which the “*” operator indicates a cross-correlation, and in        which linear dependence on φ({right arrow over (r)}) is        immediately evident.    -   For good order (and purposes of comparison), the imaging        situation in a HAADF-STEM is given by:

I ^(HAADF-STEM)({right arrow over (r)} _(p))=C _(HAADF)(|ψ_(in)({rightarrow over (r)})|²*φ²({right arrow over (r)})({right arrow over (r)}_(p))

-   -   where C_(HAADF) is a constant whose value depends on particulars        of the employed detector configuration, and “*” again indicates        a cross-correlation. It is clear that, in this technique,        imaging is a function of φ²({right arrow over (r)}) [as already        stated above].        -   One needs to make use of a special type of detector, viz a            so-called pinhole detector. The reason such a detector is            necessary is to simplify the imaging mathematics, by            allowing a rather complicated (generic) “window function” to            be replaced by a much simpler (specific) Dirac delta            function (this simplification has already been processed in            the mathematics set forth in the previous item). Such a            pinhole detector needs to be carefully manufactured (to have            a well-defined, sufficiently small pinhole) and kept            properly aligned with the imaging beam, thus introducing            extra complications relative to the present invention.

For purposes of completeness, it is noted that the “iVF documents”referred to above inter alia make it clear that:

-   -   To obtain iVF imagery, the detector in the current invention        may, for example, be a four-quadrant detector, pixelated        detector or Position-sensitive Detector (PSD), for instance.    -   To obtain iVF imagery, the vector(ized) output from such a        detector will be subjected to a two-dimensional integration        operation (e.g. Vector Field Integration or Gradient Field        Integration), thus yielding a scalar image.    -   If desired, a “raw” iVF image can be post-processed, e.g. by        applying high-, low- or band-pass filtering, Opening Angle        Correction (OAC), or deconvolution correction, for instance,        thus obtaining a so-called PiVF image.    -   If desired, one can apply a Laplacian operator to an iVF or PiVF        image, thus yielding so-called LiVF or LPiVF images,        respectively.        Any such iVF, PiVF, LiVF or LPiVF image is considered as falling        within the scope of the “integrated vector field image I” of the        current invention.

In a particular embodiment of the current invention, an approach isadopted wherein:

-   -   The specimen is conceptually sub-divided into a (depth) series        [S₁, . . . , S_(m)] of m slices disposed along and normal to        said particle-optical axis;    -   For each value of n, the corresponding image I_(n) is expressed        as a linear sum Σ_(j=1) ^(j=m)i_(n)(S_(j)) of discrete        sub-images, each associated with a different one of said slices.        For example, in a particular instance of such an embodiment, PCA        is applied to a set of m spatially aligned (and, if necessary,        scaled) iVF images acquired at different/incremental focus        values. After mean-centering each image and applying PCA, one        obtains a set of m de-correlated images that are related to the        input ones by linear transformations (i.e. each input image can        be expressed as a linear combination of these de-correlated        images). The linear mappings can be obtained using various        suitable methods, such as a so-called Karhunen-Loève Transform,        for example. New information in iVF images acquired at        increasingly deeper focus is mostly due to signals coming from        new depth layers reached by the incident charged-particle beam;        the effect of PCA de-correlation thus results in the effective        separation of the different depth layers. The inventors observed        that sets of images with lower Eigenvalues in a Karhunen-Loève        Transform correspond to deeper layers. In the image associated        with these deeper components, top layers are canceled using        information from all available shallower-focus images. Based on        these observations, one can develop an exemplary algorithm that        uses m input images, as follows:    -   Step1: Acquire m iVF images at increasingly deep focal levels        (focus series).    -   Step2: Laterally align and/or scale the image sequence thus        obtained.    -   Step3: To compute (distil) the image associated with a discrete        layer (level) of ordinal k counted from the specimen surface        (k=1 . . . m):        -   Apply PCA decomposition to the first k images in the            sequence.        -   Boost independent components having low weight (which            emanate from deeper layers); this can, for example, be done            by multiplying such components by a weighting factor that is            equal or proportional to the reciprocal of their PCA (e.g.            Karhunen-Loève) Eigenvalue.        -   Reconstruct a depth image with re-weighted independent            components (including, for example, a background (matrix or            bulk) component).

Step 4: Post-process the obtained sequence using de-noising andrestoration methods. Using such an approach, the relative thickness ofthe computed slices (layers/levels) can be adjusted by suitable choiceof the focus increments applied during acquisition of the focus series.This can result in very high depth resolution in many applications.Although the example just given makes specific use of PCA, one couldalso solve this problem using ICA or another SS technique. For moreinformation on the above-mentioned Karhunen-Loève Transform, see, forexample:

-   -   http://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%        A8ve_theorem

In a variant/special case of the embodiment set forth in the previousparagraph, the following applies:

-   -   For each given focus value P_(n), a particular slice S_(B) _(n)        is associated with a position of best focus within the specimen;    -   i_(n)(S_(j)) is set to zero for each integer j≠Bn, so that I_(n)        is taken to derive solely from S_(Bn).        In this embodiment, one is essentially assuming that a        relatively large fraction of the imaging information is        originating from the plane of best focus of the impinging        charged-particle beam, to the extent that one chooses to ignore        contributions coming from outside that plane. When viewed in        longitudinal cross-section within the specimen, such a beam has        a (quasi-) hourglass shape, with a relatively narrow “waist” (or        beam cross-over) in its middle. The charged-particle intensity        (energy per unit area) in this waist region will be maximal        compared to the rest of the beam cross-section, and what the        current embodiment essentially assumes is that imagery from this        high-intensity waist will overwhelm any imagery contributions        coming from outside the waist. The inventors have observed that        this is a reasonable assumption to make when, for example, the        beam has a relatively large opening angle (˜numerical aperture),        in which case the abovementioned hourglass will have relatively        broad (and, thus, low-intensity) “shoulders and hips” compared        to its waist. For example, an opening angle in a range greater        than about 20 mrad could be expected to give this effect.

In the context of the present invention, the set {P_(n)} (={F_(n)}) canbe referred to as a “focus series” (as already alluded to above). Theskilled artisan will understand that the cardinality of this set, andthe (incremental) separation of its elements, are matters of choice,which can be tailored at will to suit the particulars of a givensituation. In general, a larger cardinality/closer spacing of elementscan lead to higher deconvolution resolution, but will generally incur athroughput penalty. In a typical instance, one might, for example,employ a 10 cardinality of the order of about 20, with focus incrementsof the order of about 5 nm; such values are exemplary only, and shouldnot be construed as limiting.

The invention will now be elucidated in more detail on the basis ofexemplary embodiments and the accompanying schematic drawings, in which:

FIG. 1 renders a longitudinal cross-sectional elevation of an STCPM inwhich an embodiment of the current invention can be carried out.

FIG. 2 depicts a plan view of a particular embodiment of a segmenteddetector (quadrant detector) that can be used in the subject of FIG. 1,in accordance with the current invention.

FIG. 3 depicts a plan view of another embodiment of a segmented detector(pixelated detector) that can be used in the subject of FIG. 1, inaccordance with the current invention.

EMBODIMENT 1

FIG. 1 is a highly schematic depiction of an embodiment of a STCPM Maccording to the current invention, which, in this case, is a (S)TEM(though, in the context of the current invention, it could just asvalidly be an ion-based or proton microscope, for example). In theFigure, within a vacuum enclosure E, an electron source 4 (such as aSchottky emitter, for example) produces a beam (B) of electrons thattraverse an electron-optical illuminator 6, serving to direct/focus themonto a chosen part of a specimen S (which may, for example, be (locally)thinned/planarized). This illuminator 6 has an electron-optical axis B′,and will generally comprise a variety of electrostatic/magnetic lenses,(scan) deflector(s) D, correctors (such as stigmators), etc.; typically,it can also comprise a condenser system (in fact, the whole of item 6 issometimes referred to as “a condenser system”).

The specimen S is held on a specimen holder H. As here illustrated, partof this holder H (inside enclosure E) is mounted in a cradle A′ that canbe positioned/moved in multiple degrees of freedom by a positioningdevice (stage) A; for example, the cradle A′ may (inter alia) bedisplaceable in the X, Y and Z directions (see the depicted Cartesiancoordinate system), and may be rotated about a longitudinal axisparallel to X. Such movement allows different parts of the specimen S tobe irradiated/imaged/inspected by the electron beam traveling along axisB′ (and/or allows scanning motion to be performed as an alternative tobeam scanning [using deflector(s) D], and/or allows selected parts ofthe specimen S to be machined by a (non-depicted) focused ion beam, forexample).

The (focused) electron beam B traveling along axis B′ will interact withthe specimen S in such a manner as to cause various types of“stimulated” radiation to emanate from the specimen S, including (forexample) secondary electrons, backscattered electrons, X-rays andoptical radiation (cathodoluminescence). If desired, one or more ofthese radiation types can be detected with the aid of sensor 22, whichmight be a combined scintillator/photomultiplier or EDX(Energy-Dispersive X-Ray Spectroscopy) module, for instance; in such acase, an image could be constructed using basically the same principleas in a SEM. However, of principal importance in a (S)TEM, one caninstead/supplementally study electrons that traverse (pass through) thespecimen S, emerge (emanate) from it and continue to propagate(substantially, though generally with some deflection/scattering) alongaxis B′. Such a transmitted electron flux enters an imaging system(combined objective/projection lens) 24, which will generally comprise avariety of electrostatic/magnetic lenses, deflectors, correctors (suchas stigmators), etc. In normal (non-scanning) TEM mode, this imagingsystem 24 can focus the transmitted electron flux onto a fluorescentscreen 26, which, if desired, can be retracted/withdrawn (asschematically indicated by arrows 26′) so as to get it out of the way ofaxis B′. An image (or diffractogram) of (part of) the specimen S will beformed by imaging system 24 on screen 26, and this may be viewed throughviewing port 28 located in a suitable part of a wall of enclosure E. Theretraction mechanism for screen 26 may, for example, be mechanicaland/or electrical in nature, and is not depicted here.

As an alternative to viewing an image on screen 26, one can instead makeuse of the fact that the depth of focus of the electron flux emergingfrom imaging system 24 is generally quite large (e.g. of the order of 1meter). Consequently, various types of sensing device/analysis apparatuscan be used downstream of screen 26, such as:

-   -   TEM camera 30. At camera 30, the electron flux can form a static        image (or diffractogram) that can be processed by controller 10        and displayed on a display device (not depicted), such as a flat        panel display, for example. When not required, camera 30 can be        retracted/withdrawn (as schematically indicated by arrows 30′)        so as to get it out of the way of axis B′.    -   STEM detector 32. An output from detector 32 can be recorded as        a function of (X,Y) scanning position of the beam B on the        specimen S, and an image can be constructed that is a “map” of        output from detector 32 as a function of X,Y. Typically,        detector 32 will have a much higher acquisition rate (e.g. 10⁶        points per second) than camera 30 (e.g. 10² images per second).        In conventional tools, detector 32 can comprise a single pixel        with a diameter of e.g. 20 mm, as opposed to the matrix of        pixels characteristically present in camera 30; however, in the        context of the present invention, detector 32 will have a        different structure (see below), so as to allow iVF imaging to        be performed. Once again, when not required, detector 32 can be        retracted/withdrawn (as schematically indicated by arrows 32′)        so as to get it out of the way of axis B′ (although such        retraction would not be a necessity in the case of a        donut-shaped annular dark field detector 32, for example; in        such a detector, a central hole would allow beam passage when        the detector was not in use).    -   As an alternative to imaging using camera 30 or detector 32, one        can also invoke spectroscopic apparatus 34, which could be an        EELS module, for example.        It should be noted that the order/location of items 30, 32 and        34 is not strict, and many possible variations are conceivable.        For example, spectroscopic apparatus 34 can also be integrated        into the imaging system 24.

Note that the controller/computer processor 10 is connected to variousillustrated components via control lines (buses) 10′. This controller 10can provide a variety of functions, such as synchronizing actions,providing setpoints, processing signals, performing calculations, anddisplaying messages/information on a display device (not depicted).Needless to say, the (schematically depicted) controller 10 may be(partially) inside or outside the enclosure E, and may have a unitary orcomposite structure, as desired. The skilled artisan will understandthat the interior of the enclosure E does not have to be kept at astrict vacuum; for example, in a so-called “Environmental (S)TEM”, abackground atmosphere of a given gas is deliberatelyintroduced/maintained within the enclosure E. The skilled artisan willalso understand that, in practice, it may be advantageous to confine thevolume of enclosure E so that, where possible, it essentially hugs theaxis B′, taking the form of a small tube (e.g. of the order of 1 cm indiameter) through which the employed electron beam passes, but wideningout to accommodate structures such as the source 4, specimen holder H,screen 26, camera 30, detector 32, spectroscopic apparatus 34, etc.

In the context of the current invention, the following specific pointsdeserve further elucidation:

-   -   The detector 32 is embodied as a segmented detector, which, for        example, may be a quadrant sensor, pixelated CMOS/CCD/SSPM        detector, or PSD, for instance. Specific embodiments of such        detectors are shown in plan view in FIGS. 2 and 3, and will be        discussed below.    -   If a charged-particle beam propagating along the        particle-optical axis B′ traverses the specimen S without        undergoing any scattering/deflection in the specimen, then it        will impinge (substantially) symmetrically on the center/origin        O of the detector 32, and (essentially) give a “null” reading.        This situation is shown in more detail in FIGS. 2 and 3, which        show Cartesian axes X, Y with an origin at point O, on which is        centered a dashed circle that schematically represents an        impingement footprint F′ of a (ghost) charged-particle beam with        barycenter C′, such that:        -   In FIG. 2, this footprint F′ is symmetrically overlaid on            detection quadrants (electrodes) Q1, Q2, Q3, Q4. If the            detection signals (electrical currents) from these quadrants            are respectively denoted by S1, S2, S3, S4, then this            situation will yield zero difference signals S1-S3 and S2-S4            between opposing pairs of quadrants.        -   In FIG. 3, which depicts an orthogonal matrix of detection            pixels p (e.g. in a CMOS detector, possibly with an overlaid            scintillation layer), there is zero deviation between the            elected origin O of said pixel matrix and barycenter C′.    -   If, on the other hand, a charged-particle beam undergoes some        scattering/deflection in the specimen S, it will land on the        detector 32 at a position displaced from the origin O. In this        context, FIGS. 2 and 3 show a beam footprint F with barycenter C        that is no longer centered on O. The position of point C with        respect to O defines a vector V, with an associated magnitude        (length) and direction (pointing angle with respect to X axis,        for example). This vector V can be expressed in terms of the        coordinates (X_(C), Y_(C)) of point C, which can be distilled as        follows:        -   In FIG. 2, one can derive (rudimentary) estimators for            X_(C), Y_(C) using the following formulae:

$\begin{matrix}{{X_{C}\text{\textasciitilde}\frac{{S\; 1} - {S\; 3}}{{S\; 1} + {S\; 2} + {S\; 3} + {S\; 4}}},{Y_{C}\text{\textasciitilde}\frac{{S\; 2} - {S\; 4}}{{S\; 1} + {S\; 2} + {S\; 3} + {S\; 4}}}} & (1)\end{matrix}$

-   -   -   In FIG. 3, one can derive values for X_(C), Y_(C) by            examining output signals from the various pixels p, because            pixels p that are impinged upon by the beam footprint F will            give a different output signal (electrical resistance,            voltage or current, for example) to pixels p outside the            footprint F. The location of C can then be directly deduced            by noting the coordinates of that particular pixel that            yields an extremal signal, or indirectly determined by            mathematically calculating the barycenter of the cluster of            pixels p impinged on by F, or via a hybrid technique that            combines both approaches, for example.

    -   The skilled artisan will understand that the size of beam        footprint F can be altered by adjusting the so-called “camera        length” of the STCPM of FIG. 1, for example.

    -   As the input charged-particle beam B is scanned across the        specimen S so as to trace out a two-dimensional scan path        (area), the approach set forth in the previous item can be used        to obtain a value of V for each coordinate position along said        scan path. This allows compilation of a “map” of vector V as a        function of scan position on the specimen S, which amounts to a        mathematical field (and also a physical field, in that the        vector V can be assigned a (proportional) physical meaning, such        as electrostatic field vector).

    -   The vector field resulting from the previous step can now be        integrated two-dimensionally, so as to obtain an integrated        vector field (iVF) image that is intrinsic to the current        invention. This aspect of the invention will be elucidated in        more detail in the next Embodiment (which again makes specific        reference to (S)TEM, but is equally applicable to a generic        STCPM).

    -   In accordance with the present invention, such an iVF image is        obtained at each of a series {F_(n)} of different focus values        (focus position being the beam parameter P that, in accordance        with the current invention, is selected to be varied so as to        obtain input data pairs for the ensuing mathematical        deconvolution procedure; this contrasts with HAADF-STEM        tomography, for example, where the varied parameter P is chosen        to be beam tilt). Focus value may, for example, be varied by:        -   Changing a (focus) setting of (at least one of) the            particle-optical components in illuminator 6; and/or        -   Changing a Z position of specimen holder H.

    -   In this way, one accrues a measurement set M={(I_(n),        P_(n))}={(I_(n), F_(n))}, in which I_(n) is the iVF image        corresponding to a given focus value F_(n). According to the        invention, this measurement set M can then be (automatically)        deconvolved/spatially resolved into a (tomographic) result set        representing depth-resolved/depth-reconstructed imagery of the        specimen, e.g. using SS mathematical techniques as set forth        above. See, in this regard, Embodiments 3 and 4 below.

EMBODIMENT 2

A further explanation will now be given regarding some of themathematical techniques that can be used to obtain an iVF image asemployed in the present invention.

Integrating Gradient Fields As set forth above, a measured vector field{tilde over (E)}(x, y)=({tilde over (E)}_(x)(x, y), {tilde over(E)}_(y)(x, y))^(T) can (for example) be derived at each coordinatepoint (x, y) from detector segment differences using the expressions:

$\begin{matrix}{{\overset{\sim}{E}}_{x} = \frac{S_{1} - S_{3}}{S_{1} + S_{2} + S_{3} + S_{4}}} & \left( {2a} \right) \\{{\overset{\sim}{E}}_{y} = \frac{S_{2} - S_{4}}{S_{1} + S_{2} + S_{3} + S_{4}}} & \left( {2b} \right)\end{matrix}$

where, for simplicity, spatial indexing (x, y) in the scalar fields{tilde over (E)}_(x), {tilde over (E)}_(y) and S_(i=1, . . . , 4) hasbeen omitted, and where superscript T denotes the transpose of a matrix.It is known from the theory of (S)TEM contrast formation that {tildeover (E)} is a measurement of the actual electric field E in an area ofinterest of the imaged specimen. This measurement is inevitablycorrupted by noise and distortions caused by imperfections in optics,detectors, electronics, etc. From basic electromagnetism, it is knownthat the electrostatic potential function φ(x, y) [also referred tobelow as the potential map] is related to the electric field by:

E=−∇φ  (3)

The goal here is to obtain the potential map at each scanned location ofthe specimen. But the measured electric field in its noisy form {tildeover (E)} will most likely not be “integrable”, i.e. cannot be derivedfrom a smooth potential function by the gradient operator. The searchfor an estimate {circumflex over (φ)} of the potential map given thenoisy measurements {tilde over (E)} can be formulated as a fittingproblem, resulting in the functional minimization of objective functionJ defined as:

J(φ)=∫∫∥(−∇φ)−{tilde over (E)}∥ ²dxdy=∫∫∥Λφ+{tilde over (E)}∥ ²dxdy  (4)

where

${\nabla\phi} = {\left( {\phi_{x},\phi_{y}} \right)^{T} = {\left( {\frac{\partial\phi}{\partial x},\frac{\partial\phi}{\partial y}} \right)^{T}.}}$

One is essentially looking for the closest fit to the measurements, inthe least squares sense, of gradient fields derived from smoothpotential functions φ.

To be at the sought minimum of J one must satisfy the Euler-Lagrangeequation:

$\begin{matrix}{{\frac{\partial{{{\nabla\phi} + \overset{\sim}{E}}}^{2}}{\partial\phi} - {\frac{d}{dx}\frac{\partial{{{\nabla\phi} + \overset{\sim}{E}}}^{2}}{\partial\phi_{x}}} - {\frac{d}{dy}\frac{\partial{{{\nabla\phi} + \overset{\sim}{E}}}^{2}}{\partial\phi_{y}}}} = 0} & (5)\end{matrix}$

which can be expanded to:

$\begin{matrix}{{{{- \frac{d}{dx}}\frac{\partial\left\lbrack {\left( {\phi_{x} + {\overset{\sim}{E}}_{x}} \right)^{2} + \left( {\phi_{y} + {\overset{\sim}{E}}_{y}} \right)^{2}} \right\rbrack}{\partial\phi_{x}}} - {\frac{d}{dy}\frac{\partial\left\lbrack {\left( {\phi_{x} + {\overset{\sim}{E}}_{x}} \right)^{2} + \left( {\phi_{y} + {\overset{\sim}{E}}_{y}} \right)^{2}} \right\rbrack}{\partial\phi_{y}}}} = 0} & (6)\end{matrix}$

finally resulting in:

$\begin{matrix}{{\frac{\partial^{2}\phi}{\partial x^{2}} + \frac{\partial^{2}\phi}{\partial y^{2}}} = {- \left( {\frac{\partial\overset{\sim}{E}}{\partial x} + \frac{\partial\overset{\sim}{E}}{\partial y}} \right)}} & (7)\end{matrix}$

which is the Poisson equation that one needs to solve to obtain{circumflex over (φ)}.

Poisson Solvers

Using finite differences for the derivatives in (7) one obtains:

$\begin{matrix}{{\frac{\phi_{{i + 1},j} - {2\phi_{i,j}} + \phi_{{i - 1},j}}{\Delta^{2}} + \frac{\phi_{i,{j + 1}} - {2\phi_{i,j}} + \phi_{i,{j - 1}}}{\Delta^{2}}} = {- \left( {\frac{\left( {\overset{\sim}{E}}_{x} \right)_{{i + 1},j} - \left( {\overset{\sim}{E}}_{x} \right)_{{i - 1},j}}{\Delta} + \frac{\left( {\overset{\sim}{E}}_{y} \right)_{{i + 1},j} - \left( {\overset{\sim}{E}}_{y} \right)_{{i - 1},j}}{\Delta}} \right)}} & (8)\end{matrix}$

where Δ is the so-called grid step size (assumed here to be equal in thex and y directions). The right side quantity in (8) is known frommeasurements and will be lumped together in a term p_(i,j) to simplifynotation:

$\begin{matrix}{{\frac{\phi_{i + 1} - {2\phi_{i,j}} + \phi_{{i - 1},j}}{\Delta^{2}} + \frac{\phi_{i,{j + 1}} - {2\phi_{i,j}} + \phi_{i,{j - 1}}}{\Delta^{2}}} = \rho_{i,j}} & (9)\end{matrix}$

which, after rearranging, results in:

φ_(i−1,j)+φ_(i,j−1)−4φ_(i,j)+φ_(i,j+1)+φ_(i+1,j)=Δ²ρ_(i,j)  (10)

for i=2, . . . , N−1 and j=2, . . . , M−1, with (N, M) the dimensions ofthe image to be reconstructed.The system in (10) leads to the matrix formulation:

Lϕ=ρ  (11)

where ϕ and ρ represent the vector form of the potential map andmeasurements, respectively (the size of these vectors is N×M, which isthe size of the image). The so-called Laplacian matrix L is ofdimensions (N×M)² but is highly sparse and has a special form called“tridiagonal with fringes” for the discretization scheme used above.So-called Dirichlet and Neumann boundary conditions are commonly used tofix the values of {circumflex over (φ)} at the edges of the potentialmap.

The linear system of (11) tends to be very large for typical (S)TEMimages, and will generally be solved using numerical methods, such asthe bi-conjugate gradient method. Similar approaches have previouslybeen used in topography reconstruction problems, as discussed, forexample, in the journal article by Ruggero Pintus, Simona Podda andMassimo Vanzi, 14^(th) European Microscopy Congress, Aachen, Germany,pp. 597-598, Springer (2008). One should note that other forms ofdiscretization of the derivatives can be used in the previouslydescribed approach, and that the overall technique is conventionallyknown as the Poisson solver method. A specific example of such a methodis the so-called multi-grid Poisson solver, which is optimized tonumerically solve the Poisson equation starting from a coarse mesh/gridand proceeding to a finer mesh/grid, thus increasing integration speed.

Basis Function Reconstruction

Another approach to solving (7) is to use the so-called Frankot-Chellapaalgorithm, which was previously employed for depth reconstruction fromphotometric stereo images. Adapting this method to the current problem,one can reconstruct the potential map by projecting the derivatives intothe space-integrable Fourier basis functions. In practice, this is doneby applying the Fourier Transform FT(⋅) to both sides of (7) to obtain:

(ω_(x) ² +w _(y) ²)FT(φ)=√{square root over (−1)}(ω_(x) FT({tilde over(E)} _(x))+(ω_(y) FT({tilde over (E)} _(y)))  (12)

from which {circumflex over (φ)} can be obtained by Inverse FourierTransform (IFT):

$\begin{matrix}{\hat{\phi} = {{IFT}\left( {{- \sqrt{- 1}}\frac{{\omega_{x}{{FT}\left( {\overset{\sim}{E}}_{x} \right)}} + {\omega_{y}{{FT}\left( {\overset{\sim}{E}}_{y} \right)}}}{\omega_{x}^{2} + \omega_{y}^{2}}} \right)}} & (13)\end{matrix}$

The forward and inverse transforms can be implemented using theso-called Discrete Fourier Transform (DFT), in which case the assumedboundary conditions are periodic. Alternatively, one can use theso-called Discrete Sine Transform (DST), which corresponds to the use ofthe Dirichlet boundary condition (φ=0 at the boundary). One can also usethe so-called Discrete Cosine Transform (DCT), corresponding to the useof the Neumann boundary conditions (∇φ·n=0 at the boundary, n being thenormal vector at the given boundary location).

Generalizations and Improved Solutions

While working generally well, the Poisson solver and Basis Functiontechniques can be enhance further by methods that take into accountsharp discontinuities in the data (outliers). For that purpose, theobjective function J can be modified to incorporate a different residualerror R (in (4), the residual error was R(v)=∥v∥²). One can for exampleuse exponents of less than two including so-called Lp norm-basedobjective functions:

$\begin{matrix}{{= {{\int{\int{{R\left( {{- {\nabla\phi}},\overset{\sim}{E}} \right)}{dxdy}}}} = {\int{\int{{{\left( {- {\nabla\phi}} \right) - \overset{\sim}{E}}}^{\frac{1}{p}}{dxdy}}}}}},\; {p \geq 1}} & (14)\end{matrix}$

-   -   The residual can also be chosen from the set of functions        typically used in so-called M-mmonly estimations (a commonly        used class of robust estimators). In this case, R can be chosen        from among functions such as so-called Huber, Cauchy, and Tuckey        functions. Again, the desired result from this modification of        the objective function will be to avoid overly smooth        reconstructions and to account more accurately for real/physical        discontinuities in the datasets. Another way of achieving this        is to use anisotropic weighting functions w_(x) and w_(y) in J:

J(φ)=∫∫w _(x)(∈_(x) ^(k-1))(−φ_(x) −{tilde over (E)} _(x))² +w_(y)(∈_(y) ^(k-1))(−φ_(y) −{tilde over (E)} _(y))²dxdy  (15)

where the weight functions depend on the residuals:

R(∈_(x) ^(k-1))=R(φ_(x) ^(k-1) ,{tilde over (E)} _(x)) and R(∈_(y)^(R-1))=R(−φ_(y) ^(k-1) ,{tilde over (E)} _(y))  (15a)

at iteration k−1.

It can be shown that, for the problem of depth reconstruction fromphotometric stereo images, the use of such anisotropic weights, whichcan be either binary or continuous, leads to improved results in thedepth map recovery process.

In another approach, one can also apply a diffusion tensor D to thevector fields ∇φ and {tilde over (E)} with the aim of smoothing the datawhile preserving discontinuities during the process of solving for{circumflex over (φ)}, resulting in the modification of (4) into:

J(φ)=∫∫∥D(−∇φ)−D({tilde over (E)})∥²dxdy  (16)

Finally, regularization techniques can be used to restrict the solutionspace. This is generally done by adding penalty functions in theformulation of the objective criterion I such as follows:

J(φ)=∫∫[∥(−∇φ)−{tilde over (E)}∥ ²+λƒ(∇φ)]dxdy  (17)

The regularization function ƒ(∇φ) can be used to impose a variety ofconstraints on p for the purpose of stabilizing the convergence of theiterative solution. It can also be used to incorporate into theoptimization process prior knowledge about the sought potential field orother specimen/imaging conditions.

Position Sensitive Detector (PSD)

Using a Position Sensitive Detector (PSD) and measuring a thin,non-magnetic specimen, one obtains (by definition) the vector fieldimage components as components of the center of mass (COM) of theelectron intensity distribution I_(D)({right arrow over (k)}, {rightarrow over (r)}_(p)) at the detector plane:

I _(x) ^(COM)({right arrow over (r)} _(p))=∫∫_(−∞) ^(∞) k _(x) I_(D)({right arrow over (k)},{right arrow over (r)} _(p))d ² {right arrowover (k)} I _(y) ^(COM)({right arrow over (r)} _(p))=∫∫−∞∞k _(y) I_(D)({right arrow over (k)},{right arrow over (r)} _(p))d ² {right arrowover (k)}  (18)

where {right arrow over (r)}_(p) represents position of the probe(focused electron beam) impinging upon the specimen, and {right arrowover (k)}=(k_(x), k_(y)) are coordinates in the detector plane. The fullvector field image can then be formed as:

{right arrow over (I ^(COM))}({right arrow over (r)} _(p))=I _(x)^(COM)({right arrow over (r)} _(p))·{right arrow over (x)} ₀ +I _(y)^(COM)({right arrow over (r)} _(p))·{right arrow over (y)} ₀  (19)

where {right arrow over (x)}₀ and {right arrow over (y)}₀ go are unitvectors in two perpendicular directions.

The electron intensity distribution at the detector is given by:

I _(D)({right arrow over (k)},{right arrow over (r)} _(p))=|

{ψ_(in)({right arrow over (r)}−{right arrow over (r)} _(p))e^(iφ({right arrow over (r)}))}({right arrow over (k)})|²  (20)

where ψ_(in)({right arrow over (r)}−{right arrow over (r)}_(p)) is theimpinging electron wave (i.e. the probe) illuminating the specimen atposition {right arrow over (r)}_(p), and e^(iφ({right arrow over (r)}))is the transmission function of the specimen. The phase φ({right arrowover (r)}) is proportional to the specimen's inner electrostaticpotential field. Imaging φ({right arrow over (r)}) is the ultimate goalof any electron microscopy imaging technique. Expression (19) can bere-written as:

$\begin{matrix}{{\overset{\rightarrow}{I^{COM}}\left( \overset{->}{r_{p}} \right)} = {{\frac{1}{2\pi}\left( {{{\psi_{in}\left( \overset{->}{r} \right)}}^{2}\bigstar {\nabla{\phi \left( \overset{->}{r} \right)}}} \right)\left( {\overset{->}{r}}_{p} \right)} = {{- \frac{1}{2\pi}}\left( {{{\psi_{in}\left( \overset{->}{r} \right)}}^{2}\bigstar {\overset{->}{E}\left( \overset{->}{r} \right)}} \right)\left( {\overset{->}{r}}_{p} \right)}}} & (21)\end{matrix}$

where {right arrow over (E)}({right arrow over (r)})=−∇φ({right arrowover (r)}) is the inner electric field of the specimen—which is thenegative gradient of the electrostatic potential field of thespecimen—and the operator “*” denotes cross-correlation. It is evidentthat the obtained vector field image {right arrow over (I^(COM))}({rightarrow over (r)}_(p)) directly represents the inner electric field {rightarrow over (E)}({right arrow over (r)}) of the specimen. Its componentsare set forth in (18) above. Next, an integration step in accordancewith the current invention is performed, as follows:

I ^(ICOM)({right arrow over (r)} _(p))∫_(t={right arrow over (r)})_(ref) ^({right arrow over (r)}) ^(p) {right arrow over (I^(COM))}({right arrow over (r)})·d{right arrow over (l)}  (22)

using any arbitrary path l. This arbitrary path is allowed because, inthe case of non-magnetic specimens, the only field is the electricfield, which is a conservative vector field. Numerically this can beperformed in many ways (see above). Analytically it can be worked out byintroducing (21) into (22), yielding:

$\begin{matrix}{{I^{ICOM}\left( {\overset{->}{r}}_{p} \right)} = {\frac{1}{2\pi}\left( {{{\psi_{i\; n}\left( \overset{->}{r} \right)}}^{2}*{\phi \left( {\overset{->}{r}}_{p} \right)}} \right)\left( {\overset{->}{r}}_{p} \right)}} & (23)\end{matrix}$

It is clear that, with this proposed integration step, one obtains ascalar field image that directly represents φ({right arrow over (r)}),as already alluded to above.

EMBODIMENT 3

The linearity assumptions in image formation elucidated above can berepresented in the model:

Q=AI  (24)

in which:

I=(I₁, I₂, . . . , I_(N))^(T) is the set of iVF images acquired byvarying focus value;

Q=(Q₁, Q₂, . . . , Q_(N))^(T) is a set of source images that arestatistically de-correlated and that represent information coming fromdifferent depth layers (levels);

A=(α₁, α₂, . . . , α_(N))^(T) is a square matrix transforming theoriginal images into so-called principal components.

PCA decomposition obtains the factorization in equation (24) by findinga set of orthogonal components, starting with a search for the one withthe highest variance.

The first step consists in minimizing the criterion:

$\begin{matrix}{a_{1} = {\underset{{a} = 1}{\arg \; \max}E\left\{ \left( {a^{T}I} \right)^{2} \right\}}} & (25)\end{matrix}$

The next step is to subtract the found component from the originalimages, and to find the next layer with highest variance.

At iteration 1<k≤N, we find the kth row of the matrix A by solving:

$\begin{matrix}{a_{k} = {\underset{{a} = 1}{\arg \; \max}E\left\{ \left( {a^{T}\left( {I - {\sum\limits_{i = 1}^{k - 1}{w_{i}w_{i}^{T}I}}} \right)} \right)^{2} \right\}}} & (26)\end{matrix}$

It can be shown (see, for example, literature references [1] and [3]referred to above) that successive layer separation can be achieved byusing so-called Eigenvector Decomposition (EVD) of the covariance matrixΣ_(I) of the acquired images:

Σ_(I) =E{I ^(T) I}=EDE ^(T)  (27)

in which:

E is the orthogonal matrix of eigenvectors of Σ_(I);

D=diag(d₁, . . . , d_(N)) is the diagonal matrix of Eigenvalues.

The principal components can then be obtained as

Q=E ^(T) I  (28)

The Eigenvalues are directly related to the variance of the differentcomponents:

d _(i)=(var(Q _(i)))²  (29)

In cases in which noise plays a significant role, the components withlower weights (Eigenvalues) may be dominated by noise. In such asituation, the inventive method can be limited to the K (K<N) mostsignificant components. The choice to reduce the dimensionality of theimage data can be based on the cumulative energy and its ratio to thetotal energy:

$\begin{matrix}{r = \frac{\sum\limits_{i = 1}^{K}d_{i}}{\sum\limits_{i = 1}^{N}d_{i}}} & (30)\end{matrix}$

One can choose a limit for the number of employed layers K based on asuitable threshold value t. A common approach in PCA dimensionalityreduction is to select the lowest K for which one obtains r≥t. A typicalvalue for t is 0.9 (selecting components that represent 90% of the totalenergy).

Noise effects can be minimized by recombining several depth layers witha suitable weighting scheme. Additionally, re-weighting andrecombination of layers can be useful to obtain an image contrastsimilar to the original images. In the previously described PCAdecomposition, the strongest component (in terms of variance) iscommonly associated with the background (matrix) material. Adding thiscomponent to depth layers enhances the visual appearance and informationcontent of the obtained image. One can achieve the effect of boostingdeeper-lying layers, reducing noise, and rendering proper contrast byre-scaling the independent components by their variances andreconstructing the highest-energy image using the rescaled components,as follows:

Q=ED½E ^(T) I  (31)

The skilled artisan will appreciate that other choices for the linearweighting of depth layers can also be used.

EMBODIMENT 4

As an alternative to the PCA decomposition set forth above, one can alsoemploy an SS approach based on ICA. In ICA, one assumes a linear modelsimilar to (24). The main difference with PCA is that one minimizes ahigher-order statistical independence criterion (higher than thesecond-order statistics in PCA), such as so-called Mutual Information(MI):

$\begin{matrix}{{{MI}\left( {Q_{1},\ldots,\; Q_{N}} \right)} = {{\sum\limits_{i = 1}^{N}{H\left( Q_{i} \right)}} - {H(Q)}}} & (32)\end{matrix}$

With marginal entropies computed as:

$\begin{matrix}{{H(Q)} = {- {\sum\limits_{k = 1}^{S}{{P\left( {Q_{i} = q_{k}} \right)}{\log \left( {P\left( {Q_{i} = q_{k}} \right)} \right)}}}}} & (33)\end{matrix}$

and the joint entropy:

$\begin{matrix}{{H(Q)} = {- {\sum\limits_{k = 1}^{S}{{P\left( {{Q_{i} = q_{k}},\ldots \;,{Q_{N} = q_{k}}} \right)}{\log \left( {P\left( {{Q_{i} = q_{k}},\ldots \;,{Q_{N} = q_{k}}} \right)} \right)}}}}} & (34)\end{matrix}$

in which:

-   -   P(Q) is the probability distribution of the imaging quantity Q;    -   q_(k) is a possible value for said imaging quantity; and    -   S is the total number of scanned sites on the specimen (e.g. in        the case of rectangular images, this is the product of height        and width).

Other criteria—such as the so-called Infomax and Negentropy—can also beoptimized in ICA decomposition. Iterative methods—such as FastlCA—can beemployed to efficiently perform the associated depth layer separationtask. Adding more constraints to the factorization task can lead to moreaccurate reconstruction. If one adds the condition that sources (layers)render positive signals and that the mixing matrix is also positive, onemoves closer to the real physical processes underlying image formation.A layer separation method based on such assumptions may use theso-called Non-Negative Matrix Decomposition (NNMD) technique withiterative algorithms.

For more information, see, for example, literature references [1] and[2] cited above.

1. A method for imaging a specimen using a charged-particle microscope,comprising: scanning the specimen with a charged particle beam at aplurality of focal depths, each of the plurality of focal depthsincludes a plurality of scan positions; generating a vector field ateach of the plurality of focal depths based on signals received fromdifferent segments of a detector at each scan position of the focaldepth; two-dimensionally integrating the vector field at each of theplurality of focal depths to form a vector field (iVF) image at each ofthe plurality of focal depths; and deconvolving the iVF images at theplurality of focal depths into a set of depth-resolved images of thespecimen.
 2. The method of claim 1, wherein the vector field representsan electrostatic potential field gradient in the specimen.
 3. The methodof claim 1, wherein the iVF image represents an electrostatic potentialfield in the specimen.
 3. The method of claim 1, wherein the specimen isnon-magnetic.
 4. The method of claim 1, wherein each of the plurality ofiVF images is a linear sum of the set of depth-resolved images.
 5. Themethod of claim 1, wherein the deconvolution is a linear transformation.6. The method of claim 1, wherein the deconvolution is performed using aSource Separation algorithm.
 7. The method of claim 6, wherein saidSource Separation algorithm is selected from the group consistingIndependent Component Analysis, Principal Component Analysis,Non-Negative Matrix Factorization, and combinations and hybrids hereof.8. The method of claim 1, wherein the detector detects a flux of chargedparticles transmitted through the specimen.
 9. The method of claim 1,wherein the charged-particle microscope is a transmission electronmicroscope.
 10. A charged-particle microscope for imaging a specimen,comprising: a source for directing a charged particle beam towards thespecimen; deflectors for scanning the specimen at a focal depth; asegmented detector for receiving charged particles transmitted throughthe specimen; and a controller configured to: scan the specimen with thecharged particle beam at a plurality of focal depths, each of theplurality of focal depths includes a plurality of scan positions;generate a vector field at each of the plurality of focal depths basedon signals received from different segments of the detector at each scanposition of the focal depth; two-dimensionally integrate the vectorfield at each of the plurality of focal depths to form a vector field(iVF) image at each of the plurality of focal depths; and deconvolve theiVF images at the plurality of focal depths into a set of depth-resolvedimages of the specimen.
 11. The charged-particle microscope of claim 10,wherein the vector field represents an electrostatic potential fieldgradient in the specimen.
 12. The charged-particle microscope of claim10, wherein the iVF image represents an electrostatic potential field inthe specimen.
 13. The charged-particle microscope of claim 10, whereinthe specimen is non-magnetic.
 14. The charged-particle microscope ofclaim 10, wherein each of the multiple iVF images is a linear sum of theset of depth-resolved images.
 15. The charged-particle microscope ofclaim 10, wherein the deconvolution is a linear transformation.
 16. Thecharged-particle microscope of claim 10, wherein the deconvolution isperformed using a Source Separation algorithm.
 17. The charged-particlemicroscope of claim 16, wherein said Source Separation algorithm isselected from the group consisting Independent Component Analysis,Principal Component Analysis, Non-Negative Matrix Factorization, andcombinations and hybrids hereof.